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Random walk speed is a proper function on Teichmüller space

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  • Consider a closed surface $ M $ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $ M $ with a finite first moment. Corresponding to each point in the Teichmüller space of $ M $, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichmüller space of $ M $, and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gouëzel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.

    Mathematics Subject Classification: Primary: 05C81, 20F67; Secondary: 60B20, 30F45, 20E08.

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